Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 2 of 2
Full-Text Articles in Algebra
On The Continuity Of The Real Roots Of An Algebraic Equation, Melvin Henriksen, John R. Isbell
On The Continuity Of The Real Roots Of An Algebraic Equation, Melvin Henriksen, John R. Isbell
All HMC Faculty Publications and Research
It is well known that the root of an algebraic equation is a continuous multiple-valued function of its coefficients [5, p. 3]. However, it is not necessarily true that a root can be given by a continuous single-valued function. A complete solution of this problem has long been known in the case where the coefficients are themselves polynomials in a complex variable [3, chap. V]. For most purposes the concept of the Riemann surface enables one to bypass the problem. However, in the study of the ideal structure of rings of continuous functions, the general problem must be met directly. …
On The Prime Ideals Of The Ring Of Entire Functions, Melvin Henriksen
On The Prime Ideals Of The Ring Of Entire Functions, Melvin Henriksen
All HMC Faculty Publications and Research
Let R be the ring of entire functions, and let K be the complex field. In an earlier paper [6], the author investigated the ideal structure of R, particular attention being paid to the maximal ideals. In 1946, Schilling [9, Lemma 5] stated that every prime ideal of R is maximal. Recently, I. Kaplansky pointed out to the author (in conversation) that this statement is false, and constructed a non maximal prime ideal of R (see Theorem 1(a), below). The purpose of the present paper is to investigate these nonmaximal prime ideals and their residue class fields. The author is …